17th Century England – Onwards to Restoration.

In the 17^th century England, the middle class had carried forward their rebellion against absolute monarchy based on divine rights. The Parliament was the representation of this class and its fight. The men who now fought the Stuart Kings were precisely those who had profited from Tudor absolutism, which now began to irritate them. The lower middle class then split from their upper counterpart and rallied Cromwell. So far as the untitled and unmoneyed class was concerned, they stood largely by the throne, although they had as little to gain by the King as by the Parliament. The middle class was so afraid of the poor people as of the King. When the parliamentarians talked of a government based on consent, they had no intention of extending the franchise to the people; it was to be their own consent. Right to property, which they held to be sacred, meant to them the principle that the King had no right to tax them without their consent; it also meant a denial of property to the people who were poor.

Coke, who was appointed the Attorney General (and also the Chief Justice) in 1594, was attacking the divine rights of Kings and he regarded both King and the Parliament, as subject to common law which, to him, was the truly sovereign power in the land. Common law had to be interpreted by the judges. Throughout Europe, absolute state was becoming the order of the day. Louis XI had first subjugated the feudal nobility. The Reformation then enabled the monarchs to better the Church. Henry VIII had claimed jurisdiction and powers, which earlier no British King had done. To the discomfiture of Hobbes, the cursed Puritans had undone the work so artistically done by Henry VIII and the price had to be redesigned so that the fabric may be saved from total destruction in the hands of the rabble. Someone, like Thomas Hobbes agrees with Machiavelli that man is selfish and that human nature is bad but insists that the state could transfer the man into a moral being by the exercise of the master’s rod.  He is indebted to Bodin for his concept of sovereignty, but, unlike Bodin, would impose no limitations of Divine, Natural or Constitutional law on his subjects. He agrees with Grotius that, reason is the basis of law but insists that it must be sovereign’s reason alone. He modifies the Divine Right theory by discarding the divine origin of state and by giving Divine Right to the State instead to the King. Hobbes like Machiavelli, subordinated ethics and religion to politics and was the first prophet of unlimited sovereignty.

Elizabeth (1558 – 1603)

Elizabeth was the daughter of Henry VIII and Anne Boleyn. She ascended to the throne at the age of 25 in 1558 on the death of Queen Mary. She was not only regal but also human. Like her father, she had courage, determination and self-confidence. She was willful and domineering in all matters. Like her mother she was fond of pomp and pleasure. Her great virtue was that she loved the people of England and for their sake she was prepared to make any sacrifices. Her conservative mind made extreme Protestantism suspect to her, separated the foreign policy from any enthusiasm and fitted her to be the real maker of Anglicanism.

When she ascended to the throne, she had to face many difficulties. There was the danger of the civil war in the country. The orthodox Catholics regarded Elizabeth as an usurper and they were prepared to take up arms against her in order to support the cause of Mary Stuart. The Protestants were also bitter. They were determined to carry the Reformation further. There was also a danger of foreign invasion and conquest. A lot of money had been wasted in the French war during the reign of Mary Tudor. The coinage had also been debased. The credit goes to Elizabeth that she not only surmounted all these difficulties but were also able to make her country great and strong.

Religion at her time of ascension

The first great achievement of Elizabeth was her religious settlement. The policy followed by her was that she stopped the burning of the people. The “Act of Supremacy” eliminated the authority of the Pope in England and made Elizabeth the head of the Church of England. She took up the title of “Supreme Governor” and not “Supreme Head” as had been done by her father. The monasteries were dissolved and their lands were passed to the crown. A Book of Common Prayer in English was issued. Extremists among the Catholics and the Protestants were not prepared to reconcile themselves with her religious settlement. However, Elizabeth did not take any strong action against them so long as the people attended the Church. The Church settlement got a setback after 1570, when the Pope issued an excommunication and deposed Elizabeth and declared her to be longer a Queen of England. Her subjects were absolved from their allegiance to her. In 1571, the British Parliament passed an Act, which declared to be high treason for anyone in England to call the Queen a heretic, and usurper or an infidel. The puritans also attacked the Church settlement. They were utterly dissatisfied by the moderate character of it. Most of the ceremonies prescribed by the Church were considered by them to be a relic of Popery and they would like to abolish the same. However, the fact remains that, Elizabeth succeeded in attaining a large unity in the Church and that is the reason why she was successful against Spain, when the Armada attacked England in 1558.

Her foreign policy

The foreign policy of Elizabeth was essentially that of peace. Since England had been weakened in the reigns of Edward VI and Mary Tudor, it was not in her interest to fight against any foreign power in that condition. Her great danger was an invasion from France or Spain, both Catholic countries. However, Elizabeth took advantage of the bitter rivalry going on between France and Spain and was successful in playing off the one against the other. Her foreign policy approach was not dogmatic, but was guided by enlightened national interest. The reign of Elizabeth saw the beginning of English maritime activity. It brought naval supremacy to England.

Her rule in general

Elizabeth can rightly be called as one of the greatest rulers of England. No other ruler was called upon to face so many difficulties and none else faced them so boldly and successfully. She addressed in these words a deputation of both the houses of Parliament: “Though I be a woman, I’ve as good a courage answerable to my place as ever my father had. I’m your anointed Queen. I’ll never be by violence constrained to do anything. I thank God that I’d been endued with such qualities that if I were turned out of the realm in my petticoat, I were able to live in any place in Christendom.” She was the child of the Renaissance rather than that of the Reformation. Skeptical and tolerant in the age of intolerance, she was born and brought up to re-establish the Anglican Church and to evade religious war by a compromise between the Catholics and Protestants.

Tudor rule 

Tudor rule was arbitrary and autocratic in nature. It was despotism, under more or less parliamentary forms. Though despots, their rule was not a tyranny. It was popular despotism based upon the assent of the people and it was assented to because in the main it identified itself with national interests. Their rule in nature was of the dictatorship. The rising middle class called for a strong ruler and was ready to overlook his violence and unconstitutional acts if he would maintain peace and order. The policy of the Tudors was to rule with the support of the subservient Parliament. As a result, Parliament was degraded into an instrument of royal will. The Tudors had broken the power of the great nobles. Monarchs could influence elections and so secure the return of members who favoured his views. Thus, as against the sovereign, Parliament had little influence. The Tudors, however, never sought to override the authority of the Parliament. On the contrary, they encouraged the parliamentary action of the commons. Grave and momentous questions were brought before it such as the anti-papal measures, which cut off Pope’s authority in England. As a consequence, the importance of Parliament increased. The commons grew more self-reliant and were gradually fitted to shake off their tutelage to the crown. There was little friction between the Crown and the Parliament. Parliament submitted to royal guidance and the sovereign in its turn never sought to override its legislative authority.

One of the most important characteristics of the Tudor rule was the growth of the strong monarchy built upon the ruins of the feudal system. That was partly due to the decline of the power of the nobility and the invention of the gunpowder. Another important point was the broadening of the people’s minds on account of the Renaissance movement. The new spirit paved the way for the Reformation movement.

James I (1603 – 1625) 

James I ruled from 1603 to 1625. He was born in 1556 and he came to the throne after the expulsion of his mother from Scotland. When his mother was a prisoner in England, he was the King of Scotland. He did nothing to support the cause of his mother. The result was that Elizabeth accepted him as her successor to the throne of England. He had been brought up under rigid Calvinist discipline. He failed as the King of England, even though he was a man of great learning. He was so fond of “unbuttoning his royal stores of wisdom for the benefits of his subjects” that Henry IV of France called him the “wisest fool in Christendom”. He was intolerant to any criticism. He believed that Kings should have supreme authority over all. He believed that people had no right to revolt. “The state of monarchy is the supremest thing on Earth; for Kings are not only God’s lieutenants upon Earth and sit upon God’s throne, but even by God Himself, they are called Gods…it is sedition in subjects to dispute what a King may do in the height of his power. I will not be content that my power be disputed on.” 

Here is an extract from the Vicar of Bray, an old ballad:

                  To teach his flock he never missed,

                  Kings are by God appointed,

                  And damned are they, who do resist,

                  Or touch the Lord’s Anointed.

He had fixed views about politics and the status of Kings. He believed in the Divine Rights of Kings. This view was that Kings were Kings because God made them Kings and they were responsible to God alone for whatever they did and the people had no right to either find fault with them or to challenge their authority. It was this concept of monarchy that was responsible for all his troubles. James I stood for universal peace. He raised the slogan of “Beati Pacifici” (Blessed are the peace makers). His ideas of religious toleration were a cry in the wilderness. In spite of his good intentions, he was a complete failure as a king.

His relations with the Parliament

The relations between James I and the Parliament were not cordial. He believed in the Divine Rights of Kings, but the Parliament claimed certain rights on the basis of tradition, customs and evolutionary growth. Parliament based its rights and privileges on the score of History. Parliament, during the reign of James I, asserted with success its right to impeach the ministers of the King. It protested against the new impositions. It passed its law against monopolies. It asserted its rights to discuss all the affairs of the State although the King strongly protested against the claim. It failed to secure the right of meeting regularly. From these relations, it was clear that the struggle between the two had begun.

Common law

No account of the reign of James I can be complete without a reference to the common law lawyers headed by the Chief Justice Coke. In 1594, Coke was appointed Attorney General. He was a great champion of the common law. His view was that the propriety of all actions must be judged by the common law. There was no place for the Divine Rights of the Kings. The judges alone could resolve the conflicts between the prerogatives and the statutes. The view of Coke was different from that of Bacon who held that judges were lions, but lions under the throne. It was during the reign of James I that many English colonies were established beyond the seas. In 1612, the English East India Company set up its factory at Surat (Gujarat, India). Thus, the beginnings of the future British Empire in India and America were laid during the reign of James I.

Charles I (1625 – 1649)

James I died on the 27^th March 1625 and was succeeded by his second son, Charles I. Charles I loved those who were close near him, but was cold towards others. He was devoted to the Church of England and was punctual in his devotion to it. To Charles I, the Divine Right was the question of his faith, as deeply rooted as his belief in the Church. He was a bad judge of public questions and political men. He viewed them through the lens of his affections. He saw only rebellion in the critics of the Church of England. The reign of Charles I can be divided into four periods. The first four years of his reign from 1625 to 1629 covered the first period. During this period foreign wars were fought but lost and the relations between the King and the Parliament were bitter. The second period was covered by the years from 1629 to 1640. During this period, he ruled without a Parliament. The third period was covered by the years 1640 to 1642. It was also a period of short and long Parliament. The fourth period is covered by the two Civil wars (1642 – 1649). Charles I was executed in January 1649.

Relations with the Parliament 

The fundamental dispute between the King and the Parliament was that the Parliament was determined to become the sovereign of the country and was not prepared to allow the King to do whatever he wanted to do. The execution of Charles I shocked the people. The people were not in favour of Parliament going to such an extreme. The dignified behaviour of the King at the time of his execution also excited universal admiration. There was a strong reaction in favour of the monarch. Many called him the martyr who died for the Church of England. There were others who gave him the credit for having died for the laws and liberties of the English men. A few days after his death, a book entitled “Sikon Basilike” was published. It purported to give the views of the King on Government. It was felt that the dictatorship of the army was no guarantee to safeguard the popular institutions of England and the liberties of the people. The army could fight but could not reconstruct society. This execution was followed by military despotism, which was as bad as the tyranny of the King himself. The question has been asked whether the execution of Charles I was justified or not. From the legal point of view, there was no justification for trying the King as no process could be issued against the monarch. Moreover, the members of the court were partisans and did not come up to the ideal of impartiality as required by the judges. The only justification for the execution of Charles I was moral and political. Cromwell was right in saying that it was a cruel necessity. It was cruel because it was an extreme measure involving the execution of the King. It was a necessity because without it there would have been no liberty for Englishmen. Charles I was not at all prepared to accept any limitations on his powers. He was given many opportunities both by the Parliament and the army to come to reasonable compromise but he was declared dishonest by these very bodies and hence it was difficult to come to any sort of an agreement with him. The Parliament and the army always thought of him as extremely cynical and his dealings with these two constitutional bodies were inherently insincere in nature and thus no wonder such an insincere man was put to death.

Commonwealth 

The Commonwealth was established in England on January 4, 1649 by a proclamation by the Rump Parliament that “the people are, under God, the origin of all just power…that the commons of England in Parliament assembled, being chosen by and representing the people have the supreme power in this nation”. On February 5, the Rump decided that the House of Lords, being dangerous and useless, should be abolished. On February 6, it was resolved that “it hath been found by experience and this House doth declare that the office of the King in this nation and to have power thereof in any single person, is unnecessary and burdensome and dangerous to the liberty, safety and public interest of the people of this nation and thereof ought to be abolished”. On March 17 and 19, 1649, two Acts were passed by which the offices of the King and the House of Lords were abolished. Thus the House of Commons became the sole governing body of England. The chief organ of administration of the Commonwealth was the Council of the State. It was annually chosen by the Parliament. The council was concerned with the army, the navy, foreign affairs etc. In July 1649, was passed the “Treason Act”, which made it treasonable for anyone who published maliciously that “the Government is usurped or unlawful or that the Commons assembled in Parliament are not the supreme authority of this nation”. In September 1649, was passed the “Press Act” which muzzled the freedom of the press. The publication of any printed material without a license from the Government was forbidden. A special court of justice consisting of 12 judges was established to liquidate the enemies of the Commonwealth.

The members of the Parliament were Presbyterians and they insisted on imposing “certain fundamentals before a man should be free to propagate his opinions”. Cromwell was in favour of religious toleration for all except the Roman Catholics. Certain changes were introduced in the Church whereby it became less Presbyterian. It lost its autonomy and became subordinate to the State. Rump Parliament was “the first English Government to appreciate the importance of sea power”. It also was responsible for England a great sea power. The Rump Parliament also attended to foreign trade and the overseas empire of England. In 1651, was passed the “Navigation Act”. This Act was intended to strike a blow at the commercial power of Holland and no wonder it aroused the indignation of that country. The Rump had become unpopular with the army because it was a small body, which did not represent the whole nation. Moreover, many of its members were guilty of favouritism and corruption. Cromwell and his army urged the dissolution of the Parliament but the Rump refused to be dissolved. Cromwell could not tolerate the pride ambition and self-seeking of the members of the Parliament. On April 20, 1653, Cromwell himself went to the House of the Commons and turned out the members of the House and had the doors locked. The same afternoon, the Council of State also fell before military violence. After the dissolution of the Rump, Cromwell set up a new council of State which recommended that a Parliament of saints composed of 140 Godly men, 129 from England, 5 from Scotland and 6 from Ireland be summoned. This Parliament met in 1653. it is also known as Barebone’s Parliament. This Parliament was a unique one and it passed many laws like the solemnization of marriage a civil institution, public registration of births, marriages and death. Another law provided for the better custody of insanes. But this Parliament too failed. Cromwell was essentially a reluctant and an apologetic dictator. Lambert drew up a constitutional document called the “Instrument of Government”. It was the first and the last written English constitution. By this instrument, Cromwell was made Lord Protector for life with Council of State to help him. England, Scotland and Ireland were to be united in a single commonwealth with a Parliament representing the three countries. Parliament consisting of one House was to possess the legislative power and was to be elected every three years by a reformed electorate. This instrument gave Cromwell a limited monarchy for life. While the peculiarity of the English constitution was that it was flexible and unwritten, but the instrument tried to make it rigid and written.

Cromwell

Cromwell was one of the greatest figures in the History of England. He was born in 1599 at Huntington. He was a son of a country gentleman and was educated in a college in Cambridge. He emerged as a leader of his country when she was plunged in a Civil war on account of the conflict between the King and the Parliament. He not only won victories for the Parliament but also restored law and order in the country. Cromwell was the first pronounced imperialist in the history of England. His objective was to extend the power of England overseas and he did not hesitate to use all possible means to achieve that end. In his foreign policy, he showed zeal for Protestantism, but while doing so, he did not ignore the trade and commerce of his country. He was himself a Republican, but circumstances forced him to act as a military despot. He tried to govern by a system involving the division of power between himself and the Parliament. When he failed in that objective, he ruled despotically. He levied taxes without the sanction of the Parliament. He imprisoned people without trial. As a matter of fact, he set up a military tyranny. He wished for the Parliament to be supreme and did not wish to take up the title of the King. His faith in God was both a source of his strength and his weakness. In all that he did, whether good or evil, in the three kingdoms, his conviction was that God would support him in everything that he undertook. What he judged to be necessary for the present, that he thought to be predestined for the future. His victories seemed to him, not the result of the means, which he employed, but proofs that his policies were also the will of the Divine. Although he is regarded by some as the greatest patriot and by others as the greatest traitor, he was without doubt one of the greatest men of his country. He possessed military capacity of a very high order. He organized and maintained an army, which was so efficient that he did not meet with any defeat. Oliver Cromwell died on September 3,1658. When his strong hand was removed, the country was plunged into confusion. The Levellers stood for a Republic in which the common people were to rule without Lords, Priests or Lawyers. They had a very treatment from Oliver Cromwell but now they felt that they could do whatever they pleased. Another set known as the Fifth Monarchy Men was led by Harrison. They foretold the immediate end of the world. According to their reading of Daniel, the four monarchies of antiquity, Babylonia, Persia, Greece, and Rome were to be succeeded by the fifth monarchy, now at hand, the reign of Christ and his Saints.

Oliver Cromwell had named his son Richard Cromwell to succeed as the Lord Protector. He lacked political capacity and had no advisers. The gulf between Richard and the army was widening. In order to strengthen his position, Richard decided to summon the Parliament and the members of the Parliament were hostile to the army. The army did not like the law, which forbade the army officers from having political meetings. Parliament was dissolved by force in April 1659 and the Rump was recalled. Richard resigned his charge. About the period between 1649 and 1660, monarchy had gone and the House of Lords was established as “useless and dangerous”. This “freedom” was to rise to a climax of Puritan democracy, to decline by reaction into military dictatorship and at last to expire through faction. But it left a legacy. Puritanism released an energy, which called for liberty in religion and every department of life with efficiency greater than anything England had seen. It took long strides towards union with Scotland and Ireland. Its administrative machinery pointed towards the cabinet and its economic doctrine led to the capitalist Britain of the next two centuries.

Restoration

The Restoration of Charles II to the throne of England in 1660 was not merely the restoration of the King but also of Parliament, the Anglican Church, the historic law courts and the old system of local governments in the country. It is wrong to say that the monarchy that was restored was the unlimited and absolute monarchy of Charles I and James I. The “Triennial Act” of 1641 had not been repealed and it meant that the King could not carry on Government without calling a Parliament at least once in three years. Thus, the “Triennial Act” put a check on the power of the King. The result was that Charles II had to be less arbitrary and act according to the law. The very fact that Parliament had made Charles II, the King of England implied that in the last resort Parliament could also unmake him. The Divine Right of Kings to rule was practically dead. A point of conflict between the first two Stuart Kings and the Parliament was the question of taxation. Charles I had levied taxes without the consent of the Parliament. Now it was clearly understood that new taxes could be levied only with the consent of the Parliament. It is clear that although monarchy was restored, it was restored with a difference. Restoration gave to Parliament its old form and organization, which had so radically been changed during the Commonwealth period. The two Houses of the Parliament were restored and the House of Commons became more powerful than the House of Lords. Charles II never questioned the privileges of the Parliament. As a matter of fact, most of his important laws were passed through the Parliament.

Restoration of the Church

Restoration was also the restoration of the Church of England. The Parliament, which was elected in 1661 after the dissolution of the convention Parliament, was predominantly Anglican in nature. The Presbyterian element had disappeared altogether. The so-called Cavalier Parliament ended the work of the Presbyterian majority and restored the Anglican Church to its former position. The “Act of Uniformity” of 1662 provided that all clergymen and teachers were to declare their acceptance of the Anglican Prayer Book. The “Conventicle Act” of 1664 forbade under severe penalties, attendance of any public worship, which was not of Anglican form, of more than four persons, unless they belonged to the same family. The “Test Act” of 1673 provided that all civil and military officers were to take the oath of allegiance and accept the supremacy of the Church of England. In 1679, was passed the “Parliamentary Test Act” which provided that no person was to be a Member of Parliament unless he belonged to the Church of England. 1679 was also the year of Hobbes’ death. It is clear that the Anglican Church was established as a State Church, but with the difference that the headship of the Church no longer belonged to the King as a prerogative right. The leadership of the Church lay with the King-in-Parliament. As has been rightly put by Sir D. L. Keir that the restoration of monarchy in 1660 was especially a return to Government by Law. In this period, the legislative union with Scotland and Ireland was dissolved.

The reign of Charles II

During the reign of Charles II, some constitutional progress was made. The system of appropriation of supplies was established. While granting money to the King, the Parliament laid down the specific purpose for which the money was granted. The responsibility of the Ministers of Parliament was also secured to some extent. During the reign of Charles II, parliamentary parties with definite political programmes were formed and that also added to the strength of the Parliament. The passing of the “Habeas Corpus Act” in 1679 has become a cornerstone of the liberties of the people of England. The Parliament, which placed Charles II on the throne, was known as the Convention Parliament because it was summoned without a royal writ. The lands of the Royalists, which were confiscated, were restored. The Royal revenue was fixed at a fixed sum. Feudal dues and purveyance were abolished. A permanent excise tax was granted to the King as a compensation for the loss of his feudal revenues. The Convention Parliament was dissolved in 1661 and fresh elections were held. The new Parliament, which met in 1661, sat for 18 years and is known as the Cavalier Parliament. It was so called because the cavalier spirit was present among its members. This Parliament was royalist in nature in politics and Anglican in religion. It hated the Puritans and stood for the strengthening of the Church of England. It is true that during the reign of Charles II, the court was corrupt and there were pleasures all around. However, during this period, humanity and refinement spread rapidly in England. Literature, art and science, architecture and etiquette and fashions were copied from the court off Louis XIV, the grand monarch of France. One of the most notable men of this age wasIsaac Newton (1642 – 1727). There were great strides made in the disciplines of Astronomy, Physics, Chemistry and Medicine.

Advertisement

Consequentialism -X- (Pareto Efficiency) -X- Deontology

Let us check the Polity to begin with:

1. N is the set of all individuals in society.

And that which their politics concerns – the state of society.

2. S is the set of all possible information contained within society, so that a set s ∈ 2^S (2^S being the set of all possible subsets of S) contains all extant information about a particular iteration of society and will be called the state of society. S is an arbitrary topological space.

And the means by which individuals make judgements about that which their politics concerns. Their preferences over the information contained within the state of society.

3. Each individual i ∈ N has a complete and transitive preference relation ≽_i defined over a set of preference-information S_i ⊂ S such that s_i ≽ s′_i can be read “individual i prefers preference information s_i at least as much as preference-information s′_i”.

Any particular set of preference-information s_i ⊂ S_i can be thought of as the state of society as viewed by individual i. The set of preference-information for individual i is a subset of the information contained within a particular iteration of society, so s_i ⊂ s ⊂ S.

A particular state of society s is a Pareto efficient if there is no other state of society s′ for which one individual strictly prefers their preference-information s′_i ⊂ s′ to that particular state s_i ⊂ s, and the preference-information s′_j ⊂ s′ in the other state s′ is at least as preferred by every other individual j ≠ i.

4. A state s ∈ S is said to be Pareto efficient iff ∄ s′ ∈ 2^S & i ∈ N : s′_i ≻ s_i & s′_j ≽ s_j ∀ j ≠ i ∈ N.

To put it crudely, a particular state of society is Pareto efficient if no individual can be made “better off” without making another individual “worse off”. A dynamic concept which mirrors this is the concept of a Pareto improvement – whereby a change in the state of society leaves everyone at least indifferent, and at least one individual in a preferable situation.

5. A movement between two states of society, s → s′ is called a Pareto improvement iff ∃ i ∈ N : s′_i ≻ s_i & s′_j ≽ s_j ∀ j ≠ i ∈ N .

Note that this does not imply that s′ is a Pareto efficient state, because the same could potentially be said of a movement s′ → s′′. The state s′ is only a Pareto efficient state if we cannot find yet another state for which the movement to that state is a Pareto improvement. The following Theorem, demonstrates this distinction and gives an alternative definition of Pareto efficiency.

Theorem: A state s ∈ 2^S is Pareto efficient iff there is no other state s′ for which the movement s → s′ is a Pareto improvement.

If one adheres to a consequentialist political doctrine (such as classical utilitarianism) rather than a deontological doctrine (such as liberalism) in which action is guided by some categorical imperative other than consequentialism, the guide offered by Pareto improvement is the least controversial, and least politically committal criterion to decision-making one can find. Indeed if we restrict political statements to those which concern the assignation of losses, it is a-political. It makes a value judgement only about who ought gain (whosoever stands to).

Unless one holds a strict deontological doctrine in the style, say, of Robert Nozick’s Anarchy state and Utopia (in which the maintenance of individual freedom is the categorical imperative), or John Rawls’ A Theory of Justice (in which again individual freedom is the primary categorical imperative and the betterment of the “poorest” the second categorical imperative), it is more difficult to argue against implementing some decision which will cause a change of society which all individuals in society will be at worst indifferent to. Than arguing for some decision rule which will induce a change of society which some individual will find less preferable. To the rationalisitic economist it seems almost petty, certainly irrational to argue against this criterion, like those individuals who demand “fairness” in the famous “dictator” experiment rather than accept someone else becoming “better off”, and themselves no “worse off”.

Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation S_iRS_j between the two ‘individual terms’ (subjects) S_j and S_i. In a short hand notation we represent this relation by R_ij. Relations may be composed: whenever we have relations of the form R_ij, R_jl, a third transitive relation R_il emerges following the rule

R_ijR_kl = δ_jkR_il —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

S_i = ∑_j R_ij —– (2)

The individual S_i thus defined is an eigenstate of the R_ii relation

R_iiS_i = S_i —– (3)

The relations R_ii are idempotent

R²_ii = R_ii —– (4)

and they span the identity

∑_i R_ii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1_A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, R_ij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ S_i is R_ii.

R_ij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of R_ij

R_ij = |r_i⟩⟨r_j| —– (6)

where state ⟨r_i | is the dual of the state |r_i⟩ and they obey the orthonormal condition

⟨r_i |r_j⟩ = δ_ij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

∑_n|r_i⟩⟨r_i|=1 —– (8)

All relations remain satisfied if we replace the state |r_i⟩ by |ξ_i⟩ where

|ξ_i⟩ = 1/√N ∑_n |r_i⟩⟨r_n| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |r_i⟩ is derived as the sum of all its interactions with the other states. R_ij acts as a projection, transferring from one r state to another r state

R_ij |r_k⟩ = δ_jk |r_i⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Q_ij = |q_i⟩⟨q_j | —– (11)

with

Q_ij |q_k⟩ = δ_jk |q_i⟩ —– (12)

and

∑_n |q_i⟩⟨q_i| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

R_ijQ_kl = |r_i⟩⟨r_j |q_k⟩⟨q_l | = ⟨r_j |q_k⟩ S_il —– (14)

with

S_il = |r_i⟩⟨q_l | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(S_il) = ∑_n ⟨r_n |S_il |r_n⟩ = ⟨q_l |r_i⟩ —– (16)

From the above relation we deduce

Tr(R_ij) = δ_ij —– (17)

Any operator can be expressed as a linear superposition of the R_ij

A = ∑_i,j A_ijR_ij —– (18)

with

A_ij =Tr(AR_ji) —– (19)

The individual states could be redefined

|r_i⟩ → e^iφ_i |r_i⟩ —– (20)

|q_i⟩ → e^iθ_i |q_i⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨r_i |q_j⟩ changes and therefore we need an invariant formulation for the transition |r_i⟩ → |q_j⟩. This is provided by the trace of the closed operation R_iiQ_jjR_ii

Tr(R_iiQ_jjR_ii) ≡ p(q_j, r_i) = |⟨r_i |q_j⟩|² —– (22)

The completeness relation, equation (13), guarantees that p(q_j, r_i) may assume the role of a probability since

∑_j p(q_j, r_i) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

Conjuncted: Indiscernibles – Philosophical Constructibility. Thought of the Day 48.1

Conjuncted here.

“Thought is nothing other than the desire to finish with the exorbitant excess of the state” (Being and Event). Since Cantor’s theorem implies that this excess cannot be removed or reduced to the situation itself, the only way left is to take control of it. A basic, paradigmatic strategy for achieving this goal is to subject the excess to the power of language. Its essence has been expressed by Leibniz in the form of the principle of indiscernibles: there cannot exist two things whose difference cannot be marked by a describable property. In this manner, language assumes the role of a “law of being”, postulating identity, where it cannot find a difference. Meanwhile – according to Badiou – the generic truth is indiscernible: there is no property expressible in the language of set theory that characterizes elements of the generic set. Truth is beyond the power of knowledge, only the subject can support a procedure of fidelity by deciding what belongs to a truth. This key thesis is established using purely formal means, so it should be regarded as one of the peak moments of the mathematical method employed by Badiou.

Badiou composes the indiscernible out of as many as three different mathematical notions. First of all, he decides that it corresponds to the concept of the inconstructible. Later, however, he writes that “a set δ is discernible (…) if there exists (…) an explicit formula λ(x) (…) such that ‘belong to δ’ and ‘have the property expressed by λ(x)’ coincide”. Finally, at the outset of the argument designed to demonstrate the indiscernibility of truth he brings in yet another definition: “let us suppose the contrary: the discernibility of G. A formula thus exists λ(x, a₁,…, a_n) with parameters a₁…, a_n belonging to M[G] such that for an inhabitant of M[G] it defines the multiple G”. In short, discernibility is understood as:

1. constructibility
2. definability by a formula F(y) with one free variable and no parameters. In this approach, a set a is definable if there exists a formula F(y) such that b is an element of a if F(b) holds.
3. definability by a formula F (y, z₁ . . . , z_n) with parameters. This time, a set a is definable if there exists a formula F(y, z₁,…, z_n) and sets a₁,…, a_n such that after substituting z₁ = a₁,…, z_n = a_n, an element b belongs to a iff F(b, a₁,…, a_n) holds.

Even though in “Being and Event” Badiou does not explain the reasons for this variation, it clearly follows from his other writings (Alain Badiou Conditions) that he is convinced that these notions are equivalent. It should be emphasized then that this is not true: a set may be discernible in one sense, but indiscernible in another. First of all, the last definition has been included probably by mistake because it is trivial. Every set in M[G] is discernible in this sense because for every set a the formula F(y, x) defined as y belongs to x defines a after substituting x = a. Accepting this version of indiscernibility would lead to the conclusion that truth is always discernible, while Badiou claims that it is not so.

Is it not possible to choose the second option and identify discernibility with definability by a formula with no parameters? After all, this notion is most similar to the original idea of Leibniz intuitively, the formula F(y) expresses a property characterizing elements of the set defined by it. Unfortunately, this solution does not warrant indiscernibility of the generic set either. As a matter of fact, assuming that in ontology, that is, in set theory, discernibility corresponds to constructibility, Badiou is right that the generic set is necessarily indiscernible. However, constructibility is a highly technical notion, and its philosophical interpretation seems very problematic. Let us take a closer look at it.

The class of constructible sets – usually denoted by the letter L – forms a hierarchy indexed or numbered by ordinal numbers. The lowest level L₀ is simply the empty set. Assuming that some level – let us denote it by L_α – has already been

constructed, the next level L_α+1 is constructed by choosing all subsets of L that can be defined by a formula (possibly with parameters) bounded to the lower level L_α.

Bounding a formula to L_α means that its parameters must belong to L_α and that its quantifiers are restricted to elements of L_α. For instance, the formula ‘there exists z such that z is in y’ simply says that y is not empty. After bounding it to L_α this formula takes the form ‘there exists z in L_α such that z is in y’, so it says that y is not empty, and some element from L_α witnesses it. Accordingly, the set defined by it consists of precisely those sets in L_α that contain an element from L_α.

After constructing an infinite sequence of levels, the level directly above them all is simply the set of all elements constructed so far. For example, the first infinite level L_ω consists of all elements constructed on levels L₀, L₁, L₂,….

As a result of applying this inductive definition, on each level of the hierarchy all the formulas are used, so that two distinct sets may be defined by the same formula. On the other hand, only bounded formulas take part in the construction. The definition of constructibility offers too little and too much at the same time. This technical notion resembles the Leibnizian discernibility only in so far as it refers to formulas. In set theory there are more notions of this type though.

To realize difficulties involved in attempts to philosophically interpret constructibility, one may consider a slight, purely technical, extension of it. Let us also accept sets that can be defined by a formula F (y, z₁, . . . , z_n) with constructible parameters, that is, parameters coming from L. Such a step does not lead further away from the common understanding of Leibniz’s principle than constructibility itself: if parameters coming from lower levels of the hierarchy are admissible when constructing a new set, why not admit others as well, especially since this condition has no philosophical justification?

Actually, one can accept parameters coming from an even more restricted class, e.g., the class of ordinal numbers. Then we will obtain the notion of definability from ordinal numbers. This minor modification of the concept of constructibility – a relaxation of the requirement that the procedure of construction has to be restricted to lower levels of the hierarchy – results in drastic consequences.

Dialectics: Mathematico-Philosophical Sequential Quantification. Drunken Risibility.

Figure: Graphical representation of the quantification of dialectics.

A sequence S of P philosophers along a given period of time would incorporate the P most prominent and visible philosophers in that interval. The use of such a criterion to build the time-sequence for the philosophers implies in not necessarily uniform time-intervals between each pair of subsequent entries.

The set of C measurements used to characterize the philosophers define a C−dimensional feature space which will be henceforth referred to as the philosophical space. The characteristic vector v⃗_i of each philosopher i defines a respective philosophical state in the philosophical space. Given a set of P philosophers, the average state at time i, i ≤ P, is defined as

a⃗_i = 1/i ∑_k=1ⁱ v⃗_k

The opposite state of a given philosophical state v⃗_i is defined as:

r⃗_i = v⃗_i +2(a⃗_i −v⃗_i) = 2a⃗_i − v⃗_i

The opposition vector of philosophical state v⃗_i is given by D⃗_i = r⃗_i − v⃗_i. The opposition amplitude of that same state is defined as ||D⃗_i||.

An emphasis move taking place from the philosophical state v⃗_i is any displacement from v⃗_i along the direction −r⃗_i. A contrary move from the philosophical state v⃗_i is any displacement from v⃗_i along the direction r⃗_i.

Given a time-sequence S of P philosophers, the philosophical move implied by two successive philosophers i and j corresponds to the M⃗_i,j vector extending from v⃗_i to v⃗_j , i.e.

M⃗_i,j = v⃗_j – v⃗_i

In principle, an innovative or differentiated philosophical move would be such that it departs substantially from the current philosophical state v⃗_i. Decomposing innovation moves into two main subtypes: opposition and skewness.

The opposition index W_i,j of a given philosophical move M⃗_i,j is defined as

W_i,j = 〈M⃗_i,j, D⃗_i〉/  ||D⃗_i||²

This index quantifies the intensity of opposition of that respective philosophical move, in the sense of having a large projection along the vector D⃗_i. It should also be noticed that the repetition of opposition moves lead to little innovation, as it would imply in an oscillation around the average state. The skewness index s_i,j of that same philosophical move is the distance between v⃗_j and the line L defined by the vector D⃗_i, and therefore quantifies how much the new philosophical state departs from the respective opposition move. Actually, a sequence of moves with zero skewness would represent more trivial oscillations within the opposition line L_i.

We also suggest an index to quantify the dialectics between a triple of successive philosophers i, j and k. More specifically, the philosophical state v⃗_i is understood as the thesis, the state j is taken as the antithesis, with the synthesis being associated to the state v⃗_k. The hypothesis that k is the consequence, among other forces, of a dialectics between the views v⃗_i and v⃗_j can be expressed by the fact that the philosophical state v⃗_k be located near the middle line ML_i,j defined by the thesis and antithesis (i.e. the points which are at an equal distance to both v⃗_i and v⃗_j) relatively to the opposition amplitude ||D⃗_i||.

Therefore, the counter-dialectic index is defined as

ρ_i→k = d_i→k /||M⃗_i,j||

where d_i→k is the distance between the philosophical state v⃗_k and the middle-line M_Li,j between v⃗_i and v⃗_j. Note that 0 ≤ d_i→k ≤ 1. The choice of counter-dialectics instead of dialectics is justified to maintain compatibility with the use of a distance from point to line as adopted for the definition of skewness….

Infinitesimal and Differential Philosophy. Note Quote.

If difference is the ground of being qua becoming, it is not difference as contradiction (Hegel), but as infinitesimal difference (Leibniz). Accordingly, the world is an ideal continuum or transfinite totality (Fold: Leibniz and the Baroque) of compossibilities and incompossibilities analyzable into an infinity of differential relations (Desert Islands and Other Texts). As the physical world is merely composed of contiguous parts that actually divide until infinity, it finds its sufficient reason in the reciprocal determination of evanescent differences (dy/dx, i.e. the perfectly determinable ratio or intensive magnitude between indeterminate and unassignable differences that relate virtually but never actually). But what is an evanescent difference if not a speculation or fiction? Leibniz refuses to make a distinction between the ontological nature and the practical effectiveness of infinitesimals. For even if they have no actuality of their own, they are nonetheless the genetic requisites of actual things.

Moreover, infinitesimals are precisely those paradoxical means through which the finite understanding is capable of probing into the infinite. They are the elements of a logic of sense, that great logical dream of a combinatory or calculus of problems (Difference and Repetition). On the one hand, intensive magnitudes are entities that cannot be determined logically, i.e. in extension, even if they appear or are determined in sensation only in connection with already extended physical bodies. This is because in themselves they are determined at infinite speed. Is not the differential precisely this problematic entity at the limit of sensibility that exists only virtually, formally, in the realm of thought? Isn’t the differential precisely a minimum of time, which refers only to the swiftness of its fictional apprehension in thought, since it is synthesized in Aion, i.e. in a time smaller than the minimum of continuous time and hence in the interstitial realm where time takes thought instead of thought taking time?

Contrary to the Kantian critique that seeks to eliminate the duality between finite understanding and infinite understanding in order to avoid the contradictions of reason, Deleuze thus agrees with Maïmon that we shouldn’t speak of differentials as mere fictions unless they require the status of a fully actual reality in that infinite understanding. The alternative between mere fictions and actual reality is a false problem that hides the paradoxical reality of the virtual as such: real but not actual, ideal but not abstract. If Deleuze is interested in the esoteric history of differential philosophy, this is as a speculative alternative to the exoteric history of the extensional science of actual differences and to Kantian critical philosophy. It is precisely through conceptualizing intensive, differential relations that finite thought is capable of acquiring consistency without losing the infinite in which it plunges. This brings us back to Leibniz and Spinoza. As Deleuze writes about the former: no one has gone further than Leibniz in the exploration of sufficient reason [and] the element of difference and therefore [o]nly Leibniz approached the conditions of a logic of thought. Or as he argues of the latter, fictional abstractions are only a preliminary stage for thought to become more real, i.e. to produce an expressive or progressive synthesis: The introduction of a fiction may indeed help us to reach the idea of God as quickly as possible without falling into the traps of infinite regression. In Maïmon’s reinvention of the Kantian schematism as well as in the Deleuzian system of nature, the differentials are the immanent noumena that are dramatized by reciprocal determination in the complete determination of the phenomenal. Even the Kantian concept of the straight line, Deleuze emphasizes, is a dramatic synthesis or integration of an infinity of differential relations. In this way, infinitesimals constitute the distinct but obscure grounds enveloped by clear but confused effects. They are not empirical objects but objects of thought. Even if they are only known as already developed within the extensional becomings of the sensible and covered over by representational qualities, as differences they are problems that do not resemble their solutions and as such continue to insist in an enveloped, quasi-causal state.

Some content on this page was disabled on May 4, 2020 as a result of a DMCA takedown notice from Columbia University Press. You can learn more about the DMCA here:

https://en.support.wordpress.com/copyright-and-the-dmca/

Philosophy of Local Time

Let us hypothesize on the notion of local time.

Existence of temporal order: For each concrete basic thing x ∈ Θ, there exist a single ordering relation between their states ≤.

We now give a name to this ordering relation:

Denotation of temporal order: The set of lawful states of x is temporally ordered by the ≤ relation.

The above is a partial order relation: there are pairs of states that are not ordered by ≤; e.g. given an initial condition (x₀,v₀) for a moving particle, there are states (x₁,v₁) that are not visited by the particle.

Proper history: A totally order set of states of x is called a proper history of x.

The axiomatics do not guarantee the existence of a single proper history: they allow many of them, as in “The garden of forking paths”. The following axiom forbids such possibility.

Unicity of proper history: Each thing has one and only one proper history.

Arrow of time: The axiomatics describe a kind of “arrow of time”, although it is not related to irreversibility.

A proper history is also an ontological history. The parameter t ∈ M has not to be continuous. The following axiom, a very strong version of Heraclitus’ hypothesis Panta rhei, states that every thing is changing continuously:

Continuity: If the entire set of states of an ontological history is divided in two subsets h_p and h_f such that every state in h_p temporally precedes any state in h_f, then there exists one and only one state s₀ such that s₁ ≤ s₀ ≤ s₂, where s₁ ∈ h_p and s₂ ∈ h_f.

The axiom of continuity is stated in the Dedekind form.

Continuity in quantum mechanics: Although quantum mechanical “changes of state” are usually considered “instantaneous”, theory shows that probabilities change in a continuous way. The finite width of spectral lines also shows a continuous change in time.

Real representation: Given a unit change (s₀, s₁) there exists a bijection T : h ↔ R such that

h₁ = {s(τ)|τ ∈ R} —– (1)
s₀ = s(0) —– (2)
s₁ = s(1) —– (3)

Local time: The function T is called local time. The unit change (s₀, s₁) is arbitary. It defines an arbitrary “unit of local time”.

The above theory of local time has an important philosophical consequence: becoming, which is usually conceived as evolution in time, is here more fundamental than time. The latter is constructed as an emergent property of a changing (i.e. a becoming) thing.

Neo-Con Times are Non-Sociological Times

This one is brute force.

Those who cut their eye-teeth on the likes of Austrian economics ten or fifteen years back have moved far beyond the analytical limits of the utilitarian framework and the facile solution to every social problem in terms of getting Big Government off our backs and leaving it to Smith’s invisible hand to automatically harmonize the private interests of individuals with the policy interests of the State and forge organic solidarities within and between Nations through the operation of the law of comparative advantage, while the State prudently limits itself to enforcing contracts and seeing to the military defense of the Nation. They are trying to perceive those aspects of life that lie outside the field of vision of the force-economics binocular- namely, Man’s specifically and irreducibly social being.  They have come to question the view of individuals as homogeneous, interchangeable, and isolate utility-maximizing machines endowed with rights derived from a fictive “state of Nature” in which all social relations are abstracted away as a methodological first principle, held to take shape only posterior to a putative “social contract” that binds the hitherto asocial individuals together, and only through the media of self-interested economic exchange and common subjection to the coercive power of positive law. In their rebellion against the poverty of these economic and juridical abstractions, they strive to piece together the concrete existence of the individual as a social animal compelled by its very Nature, and not just rewards and punishments, to seek out and affiliate with others….

Let’s contrast this inherently sociological current in politics to its nemesis. The latter strives to efface every aforementioned dimension of social belonging and personal identity and lump every individual into one big boundless mass, differentiated only in terms of technical specialization in the capitalist production process and by a proliferation of sexual and other “identities” shorn of their social substance and freely adopted and then discarded at will by consumers as though so many shifting vagaries of fashion. The social ties of shared descent, territory, memory, language, tradition, religion, and rule are progressively delegitimized by a relentless campaign of propaganda, homogenizing consumerist monoculture, and not least of all, coercive government action including military conquest (“regime change”). Meanwhile, asabiyyah is derided as so much ridiculously obsolete superstition, dull-witted provincialism, and mental pathology that stunts and “oppresses” the individual. Each particular society is progressively stripped of its boundary-maintaining capacity and slurred into the others- and since only particular societies exist, this means that society itself is becoming an endangered species.

The Locus of Renormalization. Note Quote.

Since symmetries and the related conservation properties have a major role in physics, it is interesting to consider the paradigmatic case where symmetry changes are at the core of the analysis: critical transitions. In these state transitions, “something is not preserved”. In general, this is expressed by the fact that some symmetries are broken or new ones are obtained after the transition (symmetry changes, corresponding to state changes). At the transition, typically, there is the passage to a new “coherence structure” (a non-trivial scale symmetry); mathematically, this is described by the non-analyticity of the pertinent formal development. Consider the classical para-ferromagnetic transition: the system goes from a disordered state to sudden common orientation of spins, up to the complete ordered state of a unique orientation. Or percolation, often based on the formation of fractal structures, that is the iteration of a statistically invariant motif. Similarly for the formation of a snow flake . . . . In all these circumstances, a “new physical object of observation” is formed. Most of the current analyses deal with transitions at equilibrium; the less studied and more challenging case of far form equilibrium critical transitions may require new mathematical tools, or variants of the powerful renormalization methods. These methods change the pertinent object, yet they are based on symmetries and conservation properties such as energy or other invariants. That is, one obtains a new object, yet not necessarily new observables for the theoretical analysis. Another key mathematical aspect of renormalization is that it analyzes point-wise transitions, that is, mathematically, the physical transition is seen as happening in an isolated mathematical point (isolated with respect to the interval topology, or the topology induced by the usual measurement and the associated metrics).

One can say in full generality that a mathematical frame completely handles the determination of the object it describes as long as no strong enough singularity (i.e. relevant infinity or divergences) shows up to break this very mathematical determination. In classical statistical fields (at criticality) and in quantum field theories this leads to the necessity of using renormalization methods. The point of these methods is that when it is impossible to handle mathematically all the interaction of the system in a direct manner (because they lead to infinite quantities and therefore to no relevant account of the situation), one can still analyze parts of the interactions in a systematic manner, typically within arbitrary scale intervals. This allows us to exhibit a symmetry between partial sets of “interactions”, when the arbitrary scales are taken as a parameter.

In this situation, the intelligibility still has an “upward” flavor since renormalization is based on the stability of the equational determination when one considers a part of the interactions occurring in the system. Now, the “locus of the objectivity” is not in the description of the parts but in the stability of the equational determination when taking more and more interactions into account. This is true for critical phenomena, where the parts, atoms for example, can be objectivized outside the system and have a characteristic scale. In general, though, only scale invariance matters and the contingent choice of a fundamental (atomic) scale is irrelevant. Even worse, in quantum fields theories, the parts are not really separable from the whole (this would mean to separate an electron from the field it generates) and there is no relevant elementary scale which would allow ONE to get rid of the infinities (and again this would be quite arbitrary, since the objectivity needs the inter-scale relationship).

In short, even in physics there are situations where the whole is not the sum of the parts because the parts cannot be summed on (this is not specific to quantum fields and is also relevant for classical fields, in principle). In these situations, the intelligibility is obtained by the scale symmetry which is why fundamental scale choices are arbitrary with respect to this phenomena. This choice of the object of quantitative and objective analysis is at the core of the scientific enterprise: looking only at molecules as the only pertinent observable of life is worse than reductionist, it is against the history of physics and its audacious unifications and invention of new observables, scale invariances and even conceptual frames.

As for criticality in biology, there exists substantial empirical evidence that living organisms undergo critical transitions. These are mostly analyzed as limit situations, either never really reached by an organism or as occasional point-wise transitions. Or also, as researchers nicely claim in specific analysis: a biological system, a cell genetic regulatory networks, brain and brain slices …are “poised at criticality”. In other words, critical state transitions happen continually.

Thus, as for the pertinent observables, the phenotypes, we propose to understand evolutionary trajectories as cascades of critical transitions, thus of symmetry changes. In this perspective, one cannot pre-give, nor formally pre-define, the phase space for the biological dynamics, in contrast to what has been done for the profound mathematical frame for physics. This does not forbid a scientific analysis of life. This may just be given in different terms.

As for evolution, there is no possible equational entailment nor a causal structure of determination derived from such entailment, as in physics. The point is that these are better understood and correlated, since the work of Noether and Weyl in the last century, as symmetries in the intended equations, where they express the underlying invariants and invariant preserving transformations. No theoretical symmetries, no equations, thus no laws and no entailed causes allow the mathematical deduction of biological trajectories in pre-given phase spaces – at least not in the deep and strong sense established by the physico-mathematical theories. Observe that the robust, clear, powerful physico-mathematical sense of entailing law has been permeating all sciences, including societal ones, economics among others. If we are correct, this permeating physico-mathematical sense of entailing law must be given up for unentailed diachronic evolution in biology, in economic evolution, and cultural evolution.

As a fundamental example of symmetry change, observe that mitosis yields different proteome distributions, differences in DNA or DNA expressions, in membranes or organelles: the symmetries are not preserved. In a multi-cellular organism, each mitosis asymmetrically reconstructs a new coherent “Kantian whole”, in the sense of the physics of critical transitions: a new tissue matrix, new collagen structure, new cell-to-cell connections . . . . And we undergo millions of mitosis each minute. More, this is not “noise”: this is variability, which yields diversity, which is at the core of evolution and even of stability of an organism or an ecosystem. Organisms and ecosystems are structurally stable, also because they are Kantian wholes that permanently and non-identically reconstruct themselves: they do it in an always different, thus adaptive, way. They change the coherence structure, thus its symmetries. This reconstruction is thus random, but also not random, as it heavily depends on constraints, such as the proteins types imposed by the DNA, the relative geometric distribution of cells in embryogenesis, interactions in an organism, in a niche, but also on the opposite of constraints, the autonomy of Kantian wholes.

In the interaction with the ecosystem, the evolutionary trajectory of an organism is characterized by the co-constitution of new interfaces, i.e. new functions and organs that are the proper observables for the Darwinian analysis. And the change of a (major) function induces a change in the global Kantian whole as a coherence structure, that is it changes the internal symmetries: the fish with the new bladder will swim differently, its heart-vascular system will relevantly change ….

Organisms transform the ecosystem while transforming themselves and they can stand/do it because they have an internal preserved universe. Its stability is maintained also by slightly, yet constantly changing internal symmetries. The notion of extended criticality in biology focuses on the dynamics of symmetry changes and provides an insight into the permanent, ontogenetic and evolutionary adaptability, as long as these changes are compatible with the co-constituted Kantian whole and the ecosystem. As we said, autonomy is integrated in and regulated by constraints, with an organism itself and of an organism within an ecosystem. Autonomy makes no sense without constraints and constraints apply to an autonomous Kantian whole. So constraints shape autonomy, which in turn modifies constraints, within the margin of viability, i.e. within the limits of the interval of extended criticality. The extended critical transition proper to the biological dynamics does not allow one to prestate the symmetries and the correlated phase space.

Consider, say, a microbial ecosystem in a human. It has some 150 different microbial species in the intestinal tract. Each person’s ecosystem is unique, and tends largely to be restored following antibiotic treatment. Each of these microbes is a Kantian whole, and in ways we do not understand yet, the “community” in the intestines co-creates their worlds together, co-creating the niches by which each and all achieve, with the surrounding human tissue, a task closure that is “always” sustained even if it may change by immigration of new microbial species into the community and extinction of old species in the community. With such community membership turnover, or community assembly, the phase space of the system is undergoing continual and open ended changes. Moreover, given the rate of mutation in microbial populations, it is very likely that these microbial communities are also co-evolving with one another on a rapid time scale. Again, the phase space is continually changing as are the symmetries.

Can one have a complete description of actual and potential biological niches? If so, the description seems to be incompressible, in the sense that any linguistic description may require new names and meanings for the new unprestable functions, where functions and their names make only sense in the newly co-constructed biological and historical (linguistic) environment. Even for existing niches, short descriptions are given from a specific perspective (they are very epistemic), looking at a purpose, say. One finds out a feature in a niche, because you observe that if it goes away the intended organisms dies. In other terms, niches are compared by differences: one may not be able to prove that two niches are identical or equivalent (in supporting life), but one may show that two niches are different. Once more, there are no symmetries organizing over time these spaces and their internal relations. Mathematically, no symmetry (groups) nor (partial-) order (semigroups) organize the phase spaces of phenotypes, in contrast to physical phase spaces.

Finally, here is one of the many logical challenges posed by evolution: the circularity of the definition of niches is more than the circularity in the definitions. The “in the definitions” circularity concerns the quantities (or quantitative distributions) of given observables. Typically, a numerical function defined by recursion or by impredicative tools yields a circularity in the definition and poses no mathematical nor logical problems, in contemporary logic (this is so also for recursive definitions of metabolic cycles in biology). Similarly, a river flow, which shapes its own border, presents technical difficulties for a careful equational description of its dynamics, but no mathematical nor logical impossibility: one has to optimize a highly non linear and large action/reaction system, yielding a dynamically constructed geodetic, the river path, in perfectly known phase spaces (momentum and space or energy and time, say, as pertinent observables and variables).

The circularity “of the definitions” applies, instead, when it is impossible to prestate the phase space, so the very novel interaction (including the “boundary conditions” in the niche and the biological dynamics) co-defines new observables. The circularity then radically differs from the one in the definition, since it is at the meta-theoretical (meta-linguistic) level: which are the observables and variables to put in the equations? It is not just within prestatable yet circular equations within the theory (ordinary recursion and extended non – linear dynamics), but in the ever changing observables, the phenotypes and the biological functions in a circularly co-specified niche. Mathematically and logically, no law entails the evolution of the biosphere.

Production Function as a Growth Model

Any science is tempted by the naive attitude of describing its object of enquiry by means of input-output representations, regardless of state. Typically, microeconomics describes the behavior of firms by means of a production function:

y = f(x) —– (1)

where x ∈ R is a p×1 vector of production factors (the input) and y ∈ R is a q × 1 vector of products (the output).

Both y and x are flows expressed in terms of physical magnitudes per unit time. Thus, they may refer to both goods and services.

Clearly, (1) is independent of state. Economics knows state variables as capital, which may take the form of financial capital (the financial assets owned by a firm), physical capital (the machinery owned by a firm) and human capital (the skills of its employees). These variables should appear as arguments in (1).

This is done in the Georgescu-Roegen production function, which may be expressed as follows:

y= f(k,x) —– (2)

where k ∈ R is a m × 1 vector of capital endowments, measured in physical magnitudes. Without loss of generality, we may assume that the first m_p elements represent physical capital, the subsequent m_h elements represent human capital and the last m_f elements represent financial capital, with m_p + m_h + m_f = m.

Contrary to input and output flows, capital is a stock. Physical capital is measured by physical magnitudes such as the number of machines of a given type. Human capital is generally proxied by educational degrees. Financial capital is measured in monetary terms.

Georgescu-Roegen called the stocks of capital funds, to be contrasted to the flows of products and production factors. Thus, Georgescu-Roegen’s production function is also known as the flows-funds model.

Georgescu-Roegen’s production function is little known and seldom used, but macroeconomics often employs aggregate production functions of the following form:

Y = f(K,L) —– (3)

where Y ∈ R is aggregate income, K ∈ R is aggregate capital and L ∈ R is aggregate labor. Though this connection is never made, (3) is a special case of (2).

The examination of (3) highlighted a fundamental difficulty. In fact, general equilibrium theory requires that the remunerations of production factors are proportional to the corresponding partial derivatives of the production function. In particular, the wage must be proportional to ∂f/∂L and the interest rate must be proportional to ∂f/∂K. These partial derivatives are uniquely determined if df is an exact differential.

If the production function is (1), this translates into requiring that:

∂²f/∂x_i∂x_j = ∂²f/∂x_j∂xi ∀i, j —– (4)

which are surely satisfied because all x_i are flows so they can be easily reverted. If the production function is expressed by (2), but m = 1 the following conditions must be added to (4):

∂²f/∂k∂x_i ∂²f/∂x_i∂k ∀i —– (5)

Conditions 5 are still surely satisfied because there is only one capital good. However, if m > 1 the following conditions must be added to conditions 4:

∂²f/∂k_i∂x_j = ∂²f/∂x_j∂k_i ∀i, j —– (6)

∂²f/∂k_i∂k_j = ∂²f/∂k_j∂k_i ∀i, j —– (7)

Conditions 6 and 7 are not necessarily satisfied because each derivative depends on all stocks of capital k_i. In particular, conditions 6 and 7 do not hold if, after capital k_i has been accumulated in order to use the technique i, capital k_j is accumulated in order to use the technique j but, subsequently, production reverts to technique i. This possibility, known as reswitching of techniques, undermines the validity of general equilibrium theory.

For many years, the reswitching of techniques has been regarded as a theoretical curiosum. However, the recent resurgence of coal as a source of energy may be regarded as instances of reswitching.

Finally, it should be noted that as any input-state-output representation, (2) must be complemented by the dynamics of the state variables:

k ̇ = g ( k , x , y ) —– ( 8 )

which updates the vector k in (2) making it dependent on time. In the case of aggregate production function (3), (8) combines with (3) to constitute a growth model.